Related Experiment Video
Updated: Jun 12, 2025

Probe Type II Band Alignment in One-Dimensional Van Der Waals Heterostructures Using First-Principles Calculations
Published on: October 12, 2019
Koopman analysis of the singularly perturbed van der Pol oscillator
Natsuki Katayama1, Yoshihiko Susuki1
1Department of Electrical Engineering, Kyoto University, Kyoto 615-8510, Japan.
Abstract:
The Koopman operator framework holds promise for spectral analysis of nonlinear dynamical systems based on linear operators. Eigenvalues and eigenfunctions of the Koopman operator, the so-called Koopman eigenvalues and Koopman eigenfunctions, respectively, mirror global properties of the system's flow. In this paper, we perform the Koopman analysis of the singularly perturbed van der Pol system. First, we show the spectral signature depending on singular perturbation: how two Koopman principal eigenvalues are ordered and what distinct shapes emerge in their associated Koopman eigenfunctions. Second, we discuss the singular limit of the Koopman operator, which is derived through the concatenation of Koopman operators for the fast and slow subsystems. From the spectral properties of the Koopman operator for the singularly perturbed system and the singular limit, we suggest that the Koopman eigenfunctions inherit geometric properties of the singularly perturbed system. These results are applicable to general planar singularly perturbed systems with stable limit cycles.
Related Concept Videos
Oscillations In An LC Circuit
Oscillations about an Equilibrium Position
Van der Waals Equation
First, the attractive forces between molecules, which are stronger at higher densities and reduce the pressure, are considered by adding to the pressure a term equal to the square of the molar density multiplied by a positive coefficient a. Second, the volume...
Transfer function and Bode Plots-II
Damped Oscillations
Although friction and other non-conservative...
MO Theory and Covalent Bonding

